1 /* $OpenBSD: s_log1pl.c,v 1.3 2013/11/12 20:35:19 martynas Exp $ */ 2 3 /* 4 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net> 5 * 6 * Permission to use, copy, modify, and distribute this software for any 7 * purpose with or without fee is hereby granted, provided that the above 8 * copyright notice and this permission notice appear in all copies. 9 * 10 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES 11 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF 12 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR 13 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES 14 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN 15 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF 16 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. 17 */ 18 19 /* log1pl.c 20 * 21 * Relative error logarithm 22 * Natural logarithm of 1+x, long double precision 23 * 24 * 25 * 26 * SYNOPSIS: 27 * 28 * long double x, y, log1pl(); 29 * 30 * y = log1pl( x ); 31 * 32 * 33 * 34 * DESCRIPTION: 35 * 36 * Returns the base e (2.718...) logarithm of 1+x. 37 * 38 * The argument 1+x is separated into its exponent and fractional 39 * parts. If the exponent is between -1 and +1, the logarithm 40 * of the fraction is approximated by 41 * 42 * log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x). 43 * 44 * Otherwise, setting z = 2(x-1)/x+1), 45 * 46 * log(x) = z + z^3 P(z)/Q(z). 47 * 48 * 49 * 50 * ACCURACY: 51 * 52 * Relative error: 53 * arithmetic domain # trials peak rms 54 * IEEE -1.0, 9.0 100000 8.2e-20 2.5e-20 55 * 56 * ERROR MESSAGES: 57 * 58 * log singularity: x-1 = 0; returns -INFINITY 59 * log domain: x-1 < 0; returns NAN 60 */ 61 62 #include <math.h> 63 64 #include "math_private.h" 65 66 /* Coefficients for log(1+x) = x - x^2 / 2 + x^3 P(x)/Q(x) 67 * 1/sqrt(2) <= x < sqrt(2) 68 * Theoretical peak relative error = 2.32e-20 69 */ 70 71 static long double P[] = { 72 4.5270000862445199635215E-5L, 73 4.9854102823193375972212E-1L, 74 6.5787325942061044846969E0L, 75 2.9911919328553073277375E1L, 76 6.0949667980987787057556E1L, 77 5.7112963590585538103336E1L, 78 2.0039553499201281259648E1L, 79 }; 80 static long double Q[] = { 81 /* 1.0000000000000000000000E0,*/ 82 1.5062909083469192043167E1L, 83 8.3047565967967209469434E1L, 84 2.2176239823732856465394E2L, 85 3.0909872225312059774938E2L, 86 2.1642788614495947685003E2L, 87 6.0118660497603843919306E1L, 88 }; 89 90 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2), 91 * where z = 2(x-1)/(x+1) 92 * 1/sqrt(2) <= x < sqrt(2) 93 * Theoretical peak relative error = 6.16e-22 94 */ 95 96 static long double R[4] = { 97 1.9757429581415468984296E-3L, 98 -7.1990767473014147232598E-1L, 99 1.0777257190312272158094E1L, 100 -3.5717684488096787370998E1L, 101 }; 102 static long double S[4] = { 103 /* 1.00000000000000000000E0L,*/ 104 -2.6201045551331104417768E1L, 105 1.9361891836232102174846E2L, 106 -4.2861221385716144629696E2L, 107 }; 108 static const long double C1 = 6.9314575195312500000000E-1L; 109 static const long double C2 = 1.4286068203094172321215E-6L; 110 111 #define SQRTH 0.70710678118654752440L 112 113 long double 114 log1pl(long double xm1) 115 { 116 long double x, y, z; 117 int e; 118 119 if( isnan(xm1) ) 120 return(xm1); 121 if( xm1 == INFINITY ) 122 return(xm1); 123 if(xm1 == 0.0) 124 return(xm1); 125 126 x = xm1 + 1.0L; 127 128 /* Test for domain errors. */ 129 if( x <= 0.0L ) 130 { 131 if( x == 0.0L ) 132 return( -INFINITY ); 133 else 134 return( NAN ); 135 } 136 137 /* Separate mantissa from exponent. 138 Use frexp so that denormal numbers will be handled properly. */ 139 x = frexpl( x, &e ); 140 141 /* logarithm using log(x) = z + z^3 P(z)/Q(z), 142 where z = 2(x-1)/x+1) */ 143 if( (e > 2) || (e < -2) ) 144 { 145 if( x < SQRTH ) 146 { /* 2( 2x-1 )/( 2x+1 ) */ 147 e -= 1; 148 z = x - 0.5L; 149 y = 0.5L * z + 0.5L; 150 } 151 else 152 { /* 2 (x-1)/(x+1) */ 153 z = x - 0.5L; 154 z -= 0.5L; 155 y = 0.5L * x + 0.5L; 156 } 157 x = z / y; 158 z = x*x; 159 z = x * ( z * __polevll( z, R, 3 ) / __p1evll( z, S, 3 ) ); 160 z = z + e * C2; 161 z = z + x; 162 z = z + e * C1; 163 return( z ); 164 } 165 166 167 /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */ 168 169 if( x < SQRTH ) 170 { 171 e -= 1; 172 if (e != 0) 173 x = 2.0 * x - 1.0L; 174 else 175 x = xm1; 176 } 177 else 178 { 179 if (e != 0) 180 x = x - 1.0L; 181 else 182 x = xm1; 183 } 184 z = x*x; 185 y = x * ( z * __polevll( x, P, 6 ) / __p1evll( x, Q, 6 ) ); 186 y = y + e * C2; 187 z = y - 0.5 * z; 188 z = z + x; 189 z = z + e * C1; 190 return( z ); 191 } 192